cp's OEIS Frontend

A114830 Each term is previous term plus ceiling of geometric mean of all previous terms.

%I A114830 #21 Feb 16 2025 08:33:00
%S A114830 1,2,4,6,9,13,18,24,31,39,48,59,71,85,101,119,139,162,187,215,246,280,
%T A114830 318,359,404,453,507,565,628,697,771,851,937,1029,1128,1234,1348,1470,
%U A114830 1600,1738,1885,2042,2209,2386,2574,2773,2984,3207,3443,3692,3955,4232,4524,4831,5154,5494,5851,6226,6620
%N A114830 Each term is previous term plus ceiling of geometric mean of all previous terms.
%C A114830 What is this sequence, asymptotically?
%H A114830 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/GeometricMean.html">Geometric Mean</a>.
%F A114830 a(1) = 1, a(n+1) = a(n) + ceiling(GeometricMean[a(1),a(2),...,a(n)]).
%F A114830 a(n+1) = a(n) + ceiling((Product_{k=1..n} a(k))^(1/n)).
%e A114830 a(2) = 1 + ceiling(1^(1/1)) = 1 + 1 = 2.
%e A114830 a(3) = 2 + ceiling[(1*2)^(1/2)] = 2 + ceiling[sqrt(2)] = 2 + 2 = 4.
%e A114830 a(4) = 4 + ceiling[(1*2*4)^(1/3)] = 4 + ceiling[CubeRoot(8)] = 4 + 2 = 6.
%e A114830 a(5) = 6 + ceiling[(1*2*4*6)^(1/4)] = 6 + floor[4thRoot(48)] = 6 + 3 = 9.
%e A114830 a(6) = 9 + ceiling[(1*2*4*6*9)^(1/5)] = 9 + ceiling[5thRoot(432)] = 9 + 4 = 13.
%e A114830 a(7) = 13 + ceiling[(1*2*4*6*9*13)^(1/6)] = 6 + floor[6thRoot(5616)] = 13 + 5 = 18.
%e A114830 a(25) = 359 + ceiling[(1 * 2 * 4 * 6 * 9 * 13 * 18 * 24 * 31 * 39 * 48 * 59 * 71 * 85 * 101 * 119 * 139 * 162 * 187 * 215 * 246 * 280 * 318 * 359)^(1/24)] = 359 + ceiling[44.8074289] = 359 + 45 = 404.
%p A114830 A114830 := proc(n)
%p A114830     option remember;
%p A114830     if n= 1 then
%p A114830         1;
%p A114830     else
%p A114830         mul(procname(i),i=1..n-1) ;
%p A114830         procname(n-1)+ceil(root[n-1](%)) ;
%p A114830     end if;
%p A114830 end proc:
%p A114830 seq(A114830(n),n=1..60) ; # _R. J. Mathar_, Jun 23 2014
%t A114830 Nest[Append[#,Last[#]+Ceiling@GeometricMean[#]]&,{1},58] (* _James C. McMahon_, Aug 20 2024 *)
%Y A114830 Cf. A065094, A065095.
%K A114830 easy,nonn
%O A114830 1,2
%A A114830 _Jonathan Vos Post_, Feb 19 2006